Remark 8.1.3.4 (03KR): Symmetry

Remark 8.1.3.4 (Symmetry). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\sigma $ be an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a morphism of simplicial sets $\operatorname{Tw}( \Delta ^{n} ) \rightarrow \operatorname{\mathcal{C}}$. Composing with the automorphism

\[ \operatorname{Tw}( \Delta ^{n} ) \xrightarrow {\sim } \operatorname{Tw}( \Delta ^{n} ) \quad \quad (i,j) \mapsto (n-j, n-i), \]

we obtain a new $n$-simplex $\overline{\sigma }$ of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. The construction $\sigma \mapsto \overline{\sigma }$ determines an isomorphism of simplicial sets $\tau : \operatorname{Cospan}(\operatorname{\mathcal{C}}) \simeq \operatorname{Cospan}(\operatorname{\mathcal{C}})^{\operatorname{op}}$, which can be described concretely as follows:

For every vertex $X \in \operatorname{\mathcal{C}}$, the morphism $\tau $ carries $X$ (regarded as a vertex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$) to itself.
Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: X \rightarrow Y$ be an edge of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, given by a pair of edges $(f: X \rightarrow B, g: Y \rightarrow B)$ of $\operatorname{\mathcal{C}}$. Then $\tau (e): Y \rightarrow X$ is the edge of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ given by the pair $(g,f)$.

Note that $\tau $ is involutive: that is, the composition

\[ \operatorname{Cospan}(\operatorname{\mathcal{C}}) \xrightarrow { \tau } \operatorname{Cospan}(\operatorname{\mathcal{C}})^{\operatorname{op}} \xrightarrow { \tau ^{\operatorname{op}} } \operatorname{Cospan}(\operatorname{\mathcal{C}}) \]

is the identity automorphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.